Galois coinvariants of the unramified Iwasawa modules of multiple $\mathbb{Z}_p$-extensions

Abstract

For a CM-field K and an odd prime number p, let K be a certain multiple Z pextension of K. In this paper, we study several basic properties of the unramified Iwasawa module X K of K as a Z p [[Gal( K /K)]]-module. Our first main result is a description of the order of a Galois coinvariant of X K in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic Z p -extension of K under a certain assumption on the splitting of primes above p. Second one is that if K is an imaginary quadratic field and p does not split in K, we give a necessary and sufficient condition for which X K is Z p [[Gal( K/K)]]-cyclic under several assumptions on the Iwasawa λ-invariant and the ideal class group of K, where K is the Z 2 p -extension of K.